Paragraph: Consider the lines: $L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2:…
Paragraph:
Consider the lines: $L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}$ Question:
The shortest distance between $L_1$ and $L_2$ is
0
$17 / \sqrt{3}$
$41 / 5 \sqrt{3}$
$17 / 5 \sqrt{3}$
Solution
The shortest distance between $L_1$ and $L_2$ is
$
\begin{array}{r}
\left|\frac{\{(2-(-1)) \hat{\mathbf{i}}+(2-2) \hat{\mathbf{j}}+(3-(-1)) \hat{\mathbf{k}}\} \cdot(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})}{5 \sqrt{3}}\right| \\
=\left|\frac{(3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}) \cdot(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})}{5 \sqrt{3}}\right|=\frac{17}{5 \sqrt{3}}
\end{array}
$